Works fine and my server properly responds to example.com:80 HTTP requests. Now I'm going to add an alias to this domain. I want all requests to my-new-domain.com:80/test produce the same responses as to example.com, for example:

my-new-domain.com/test/xyz -> example.com/xyz

What is important, I don't want to see HTTP redirections (301 code). I want to hide example.com URL from a user. All that he has to see is my-new-domain.com/test URL.

Use mod_rewrite with mod_proxy (https://httpd.apache.org/docs/2.2/mod/mod_rewrite.html). Otherwise, if domains differ, mod_rewrite performs an external rewrite even if both are served from the same host.